In what different terms can Coxeter systems be described?

Question

My starting point is this question: https://mathoverflow.net/questions/214569 As I understand it they say, that the Coxeter matrix is not sufficient to describe the group.

I thought that up to isomorphism there is a one-to-one correspondence between Coxeter matrices and Coxeter systems. Or more formal:

Say $(W,S)$ is a Coxeter system determined by a Coxeter matrix $m$. Let $s $ and $s ′$ be distinct elements of $S$. Then:

(i) (The classes of) $s$ and $s′$ are distinct in $W$.

(ii) The order of $ss′$ in $W$ is $m(s,s′)$

The Coxeter matrix defines also the Coxeter graph. So what am I missing? Can someone explain the answers in https://mathoverflow.net/questions/214569

What is the relationship between generators and the matrix?

Answer 1

A Coxeter matrix defines a Coxeter system which defines a Coxeter group.

However, in the question posed in

https://mathoverflow.net/questions/214569

one starts with a group $G$ generated by elements, say $\{s_i \mid i \in I\}$, of order two and the orders of the products $s_is_j$ are given to be entries of a Coxeter matrix, $M = (m_{ij})$. Consequently, there is a surjective homomorphism $W \to G$, where $W$ is the Coxeter group defined by the given Coxeter matrix. But this homomorphism need not be injective.

Lee Mosher gives a concrete example in his answer to this question. Here is some additional detail for his answer. The group $W$ is defined by the Coxeter system $S = \{x,y,z\}$ such that each of $xy$, $yz$, and $zx$ have order 3. This group is isomorphic to the group of isometries of the Euclidean plane generated by reflections (corresponding to $x$, $y$, and $z$) in the sides of a given equilateral triangle. Identify $W$ with this subgroup of the isometry group of the Euclidean plane.

Let $N = W \cap T$, where $T$ consists all translations. Since $T$ is normal in the isometry group (a conjugate of a translation is a translation) of the Euclidean plane, $N$ is normal in $W$.

It is clear that $xN$, $yN$, and $zN$ generate $W/N$ and have order two. Their products, $xyN$, etc. cannot have order smaller than 3 because $(xy)^2$ is not a translation, and so $xyN$, etc. have order 3 in $W/N$.

Thus, $W \to W/N = G$ is a surjection to a group $G$ generated by involutions such that the product of any two of these involutions has order 3.